This article is a newer version of an article that deals with both "love" and "hate," which can be found here.
The Greek verb for "hate" is miseo. This word doesn't have the strong emotional sense that it does in English. It describes a relationship. The word expresses a negative preference, not a strong passion. Both of the common Greek word translated as "love," agape and phileo (see this article) are used at the opposite of the word translated as "hate."
Miseo means "to hate" and in passive, "to be hated." As the opposite of the two common words translated as "love," it means not being devoted to someone, and it also means not liking them.
This verb works more like our ideas of "having an aversion" or "disliking" in English. It means something is valued much less than something else. If nothing else specific is specified, it means valued much less than all similar things.
In a sense, this "hate" is like we use the word to describe our opinion about things rather than people. When comparing two things where you dislike something relative to another thing, you might say, "I hate Chinese food but love Italian". However, this doesn't mean that you are trying to rid the world of Chinese food. It means simply that you would prefer not to eat it.
When not compared to something specific, the aversion of the Greek "hate" becomes more extreme. "I don't care if he lives or dies" captures the sense of this word.
"Hate" in the sense of miseo is not quite the opposite of either agape or phileo, but when these two concepts are contrasted, Jesus uses miseo and agapao. Conceptually, "hate" is more like the opposite of "caring" because it expresses indifference so it is the opposite of agape rather than phileo. Correctly translating these terms makes this easier to understand. For example, using the common translation for these word in the Gospels, you can "hate" and "love" at the same time. For example, you can "love" fattening foods and at the same time you "hate" fattening foods. However, this is easier to understand how you say that you like things but don't really care for them. However, using an earlier example, you cannot "hate" your mother-in-law" at the same time you "love" her.